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common algebra will never lead to a contradiction. That step
was taken only in our own generation by the German workers
in the foundations of mathematics. In this connexion it must be
kept in mind that algebra deals only -with finite processes; when
infinite processes enter, as for example in summing an infinite
series, we are thrust out of algebra into another domain. This
is emphasized because the usual elementary text labelled
"Algebra' contains a great deal- infinite geometric progressions,
for instance - that is not algebra hi the modern meaning of the
The nature of what Hamilton did in his creation of quater-
nions will show up more clearly against the background of a set
of postulates (taken from L. E. Dickson's Algebras and Their
Arithmetics. Chicago, 1023) for common algebra or, as it is
technically called, & field (English writers sometimes use corpus
as the equivalent of the German Korper or French corps).
4 A field F is a system consisting of a set S of elements a, b,
ct ... and two operations, called addition and multiplication,
which may be performed upon any two (equal or distinct)
elements a and I of S9 taken in that order, to produce uniquely
determined elements a 3 b and a O b of S, such that postulates
I-Y are satisfied. For simplicity we shall write a -f b for a  &,
and ab for a C b, and call them the sum and jwoduct, respec-
tively, of a and b. Moreover, elements of S will be called
elements of F.
"L If a and 6 are any two elements of F, a -f b and ab
are uniquely determined elements of F, and
6 -r a = a -f &,   ba = ab.
*II. If a,b*e are any three elements of F,
(a - b) - c = a -r (6 -f c),  (ab)C = a(bc),  a(b + c) = ab + ac.
*III* There exist in F two distinct elements, denoted by 0,1,
such that if n is any element of F, a -j- 0 = a, al = a (whence
0 -r a  , la  rt, by I).
*IV. Whatever be the element  of F, there exists in J1 an
element jn such that a - x = 0 (whence x -f- a = 0 by I).
*V. Whatever be the element a (distinct from 0) of F, there